Singular

As soon as one calls groebner or std in a ring with inexact coefficients
(like real or complex),
most probably the result will be unusable.

Previously, the declared ring is
[code]
ring r1 = real, (max, may, maz, mbx, mby, mbz, mcx, mcy, mcz), lp;
[/code]

It seems like there are something wrong when the ring is declared with real type. I switch to the following ring
[code]
ring r1 = 0, (max, may, maz, mbx, mby, mbz, mcx, mcy, mcz), lp;
[/code]
and convert all floating point numbers to rational form. The solver gives solution similar to Maxima now.

The full working code is as follows:
[code]
ring r1 = 0, (max, may, maz, mbx, mby, mbz, mcx, mcy, mcz), lp;

number rax, ray, raz = 2318/287, -1459/253, 2499/743;
number rbx, rby, rbz = 2698/413, -971/157, 837/187;
number rcx, rcy, rcz = 8722/1111, -1249/300, 6501/1532;
number dax, day, daz = 1139/113, -1781/684, 223/659;
number dbx, dby, dbz = 10166/1191, -3200/1059, 695/479;
number dcx, dcy, dcz = 3163/321, -2994/2993, 686/563;
number la = 1453/280;
number lb = 1972/919;
number lc = 1836/821;
number ta = 6311/1318;
number tb = 1283/1371;
number tc = 1450/341;
number fab = 491/5266;
number fac = 571/1115;
number fbc = 2042/1923;

poly f1 = raz*maz+ray*may+rax*max - la;
poly f2 = daz*maz+day*may+dax*max - ta;
poly f3 = rbz*mbz+rby*mby+rbx*mbx - lb;
poly f4 = dbz*mbz+dby*mby+dbx*mbx - tb;
poly f5 = rcz*mcz+rcy*mcy+rcx*mcx - lc;
poly f6 = dcz*mcz+dcy*mcy+dcx*mcx - tc;
poly f7 = -raz*rbz*maz*mbz-raz*rby*may*mbz-raz*rbx*max*mbz+la*raz*mbz
-ray*rbz*maz*mby-ray*rby*may*mby-ray*rbx*max*mby
+la*ray*mby-rax*rbz*maz*mbx-rax*rby*may*mbx
-rax*rbx*max*mbx+la*rax*mbx+lb*rbz*maz+lb*rby*may
+lb*rbx*max-la*lb+fab
;
poly f8 = -raz*rcz*maz*mcz-raz*rcy*may*mcz-raz*rcx*max*mcz+la*raz*mcz
-ray*rcz*maz*mcy-ray*rcy*may*mcy-ray*rcx*max*mcy
+la*ray*mcy-rax*rcz*maz*mcx-rax*rcy*may*mcx
-rax*rcx*max*mcx+la*rax*mcx+lc*rcz*maz+lc*rcy*may
+lc*rcx*max-la*lc+fac
;
poly f9 = -rbz*rcz*mbz*mcz-rbz*rcy*mby*mcz-rbz*rcx*mbx*mcz+lb*rbz*mcz
-rby*rcz*mbz*mcy-rby*rcy*mby*mcy-rby*rcx*mbx*mcy
+lb*rby*mcy-rbx*rcz*mbz*mcx-rbx*rcy*mby*mcx
-rbx*rcx*mbx*mcx+lb*rbx*mcx+lc*rcz*mbz+lc*rcy*mby
+lc*rcx*mbx-lb*lc+fbc
;

ideal I = f1, f2, f3, f4, f5, f6, f7, f8, f9;
def s = solve(I, 12);
[/code]

Could anybody help explain why I cannot define a ring with [i]real[/i] data type to use with the [i]solve[/i] function? Thanks.

Hi,

I've a system of linear-quadratic equations to solve. I tried Maxima and it gives me the correct solution. Singular gives me a complex solution, which is not accurate. Below is my code and outputs in Maxima and Singular.

Singular:
[code]
number rax, ray, raz = 8.076651, -5.766802, 3.363392;
number rbx, rby, rbz = 6.532685, -6.184719, 4.475940;
number rcx, rcy, rcz = 7.850585, -4.163334, 4.243473;
number dax, day, daz = 10.079651, -2.603802, 0.338392;
number dbx, dby, dbz = 8.535685, -3.021719, 1.450940;
number dcx, dcy, dcz = 9.853585, -1.000334, 1.218473;
number la = 5.189281;
number lb = 2.145810;
number lc = 2.236297;
number ta = 4.788316;
number tb = 0.935814;
number tc = 4.252196;
number fab = 0.093240;
number fac = 0.512108;
number fbc = 1.061882;

poly f1 = raz*maz+ray*may+rax*max - la;
poly f2 = daz*maz+day*may+dax*max - ta;
poly f3 = rbz*mbz+rby*mby+rbx*mbx - lb;
poly f4 = dbz*mbz+dby*mby+dbx*mbx - tb;
poly f5 = rcz*mcz+rcy*mcy+rcx*mcx - lc;
poly f6 = dcz*mcz+dcy*mcy+dcx*mcx - tc;
poly f7 = -raz*rbz*maz*mbz-raz*rby*may*mbz-raz*rbx*max*mbz+la*raz*mbz
-ray*rbz*maz*mby-ray*rby*may*mby-ray*rbx*max*mby
+la*ray*mby-rax*rbz*maz*mbx-rax*rby*may*mbx
-rax*rbx*max*mbx+la*rax*mbx+lb*rbz*maz+lb*rby*may
+lb*rbx*max-la*lb+fab
;
poly f8 = -raz*rcz*maz*mcz-raz*rcy*may*mcz-raz*rcx*max*mcz+la*raz*mcz
-ray*rcz*maz*mcy-ray*rcy*may*mcy-ray*rcx*max*mcy
+la*ray*mcy-rax*rcz*maz*mcx-rax*rcy*may*mcx
-rax*rcx*max*mcx+la*rax*mcx+lc*rcz*maz+lc*rcy*may
+lc*rcx*max-la*lc+fac
;
poly f9 = -rbz*rcz*mbz*mcz-rbz*rcy*mby*mcz-rbz*rcx*mbx*mcz+lb*rbz*mcz
-rby*rcz*mbz*mcy-rby*rcy*mby*mcy-rby*rcx*mbx*mcy
+lb*rby*mcy-rbx*rcz*mbz*mcx-rbx*rcy*mby*mcx
-rbx*rcx*mbx*mcx+lb*rbx*mcx+lc*rcz*mbz+lc*rcy*mby
+lc*rcx*mbx-lb*lc+fbc
;

ideal I = f1, f2, f3, f4, f5, f6, f7, f8, f9;
ideal G = groebner(I);
dim(G);
vdim(G);
G;

def s = solve(I, 12);
[/code]

Maxima:
[code]
ra : [8.076651, -5.766802, 3.363392];
rb : [6.532685, -6.184719, 4.475940];
rc : [7.850585, -4.163334, 4.243473];
da : [10.079651, -2.603802, 0.338392];
db : [8.535685, -3.021719, 1.450940];
dc : [9.853585, -1.000334, 1.218473];
la : 5.189281;
lb : 2.145810;
lc : 2.236297;
ta : 4.788316;
tb : 0.935814;
tc : 4.252196;
fab : 0.093240;
fac : 0.512108;
fbc : 1.061882;


ma : [max, may, maz];
mb : [mbx, mby, mbz];
mc : [mcx, mcy, mcz];

Hab : expand((ma . rb - la) * (lb - mb . ra));
Hac : expand((ma . rc - la) * (lc - mc . ra));
Hbc : expand((mb . rc - lb) * (lc - mc . rb));

eqns : [ma . ra = la, ma . da = ta, mb . rb = lb, mb . db = tb, mc . rc = lc, mc . dc = tc, Hab + fab = 0, Hac + fac = 0, Hbc + fbc = 0];
eqns;
sols : solve(eqns, [max, may, maz, mbx, mby, mbz, mcx, mcy, mcz]);

solsf : bfloat(sols);
[/code]

The ground truth solution is
[code]
ma: 0.397328 -0.288674, 0.093798
mb: -0.007418 -0.283056, 0.099119
mc: 0.459667 0.599035, 0.264318
[/code]

Output of Singular:
[code]
[1]:
[1]:
(0.396501063225-i*0.0585142482085)
[2]:
(-0.292459889853-i*0.267965150227)
[3]:
(0.0892914674573-i*0.318934797814)
[4]:
(-0.0228603746306-i*0.0180139060626)
[5]:
(-0.380524257884-i*0.113697415565)
[6]:
(-0.0130221380534-i*0.130812174086)
[7]:
(0.467285747848+i*0.00508868957766)
[8]:
(0.302313482452-i*0.198182075022)
[9]:
(-0.0408951640129-i*0.20385360867)
[2]:
[1]:
(0.396501063225+i*0.0585142482085)
[2]:
(-0.292459889853+i*0.267965150227)
[3]:
(0.0892914674573+i*0.318934797814)
[4]:
(-0.0228603746306+i*0.0180139060626)
[5]:
(-0.380524257884+i*0.113697415565)
[6]:
(-0.0130221380534+i*0.130812174086)
[7]:
(0.467285747848-i*0.00508868957766)
[8]:
(0.302313482452+i*0.198182075022)
[9]:
(-0.0408951640129+i*0.20385360867)
[/code]

Output of Maxima:
[code]
[[max = 3.463461b-1,may = -5.2214398b-1,maz = -1.8408099b-1,
mbx = -2.3113602b-2,mby = -3.8212235b-1,mbz = -1.4860691b-2,
mcx = 4.6593653b-1,mcy = 3.5486107b-1,mcz = 1.3156148b-2],

[max = 3.973295b-1,may = -2.8866626b-1,maz = 9.3806532b-2,
mbx = -7.4168349b-3,mby = -2.8304991b-1,mbz = 9.9125014b-2,
mcx = 4.5966645b-1,mcy = 5.9905295b-1,mcz = 2.6433629b-1]]
[/code]

Did I do something wrong in Singular? Thanks for your help.